If $PV_{t, T}(\text{Divs}) \ge K\big(1-e^{-r(T-t)}\big)$, since $P_{Eur}(S_t, K, T-t) >0$, the identity
\begin{align*}
C_{Eur}(S_t, K, T-t) = P_{Eur}(S_t, K, T-t) + (S_t-K) -PV_{t, T}(\text{Divs}) +K\big(1-e^{-r(T-t)}\big),
\end{align*}
implies that
\begin{align*}
C_{Eur}(S_t, K, T-t) > (S_t-K).
\end{align*}
That is, it is not rationale to exercise the option at time $t$ in this case. Note that the conclusion also depends on the statement that "The put must be worth at least zero", which you should also highlight.
$$ $$
Note that, it may be rationale to exercise the option at $t$, instead of the maturity $T$, only if
\begin{align*}
C_{Eur}(S_t, K, T-t) \le (S_t-K).
\end{align*}
From the above identity, this is equivalent to
\begin{align*}
P_{Eur}(S_t, K, T-t) -PV_{t, T}(\text{Divs}) +K\big(1-e^{-r(T-t)}\big) \le 0.
\end{align*}
That is, the put option price is indeed taking into consideration. For the Quiz you provided, since
\begin{align*}
P_{Eur}(S_t, K, T-t) -PV_{t, T}(\text{Divs}) +K\big(1-e^{-r(T-t)}\big) &=
0.82 -2.9851 +1.4049 \\
&< 0,
\end{align*}
it may be rationale to exercise this option early.
$$$$
Here, we say that it may be rationale to exercise early. However, this does not mean it is optimal to exercise at time $t$, as we only considered two possible exercise times $t$ and $T$. In general, we need to take all possible stopping times, $\tau$, ranged from $t$ to $T$ into consideration. That is, we need to solve an optimal stopping problem.
